TIMOSHENKO - Strength of Materials-Vol2

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STRENGTH OF MATERIALS

PART

II

Advanced Theory and Problems

BY

S. TIMOSHENKO

Professor Emeritus of Engineering Mechanics

Stanford University

THIRD EDITION

D. VAN NOSTRAND COMPANY,

INC.

PRINCETON, NEW JERSEY

TOROXTO LONDON

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D. VAN NOSTRAND COMPANY, INC. 120 Alexander St., Princeton, New Jersey 2.57 Fourth Avenue, New York 10, New York

25 Hollinger Rd., Toronto 16, Canada

Macmillan & Co., Ltd., St. Martin’s St., London, W.C. 2, England All correspondence should de addressed to the

principal ofice of the company at Princeton, N. J.

Copyright, 0, 1930, 1941, 1956 by

D. V4N NOSTRAND COMPANY, INC.

Published simultaneously in Canada by D. VAN NOSTRAND COMPANY (Canada), LTD.

All Rigfits Reserved

This book, or any parts thereof, may not be reproduced in any form without written per- mission from the author and the publishers.

Library of Congress Catalogue Card No. 55-6497

First Published, June 1930

Two Reprintings

Second Ediiion, August 1941

Fourteen Reprintings

Third Edition, March 1956

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PREFACE TO THE THIRD EDITION

In preparing the latest edition of this book, a considerable amount of new material has been added. Throughout the text, the latest references have been inserted, as well as new problems for solution and additional figures. The major changes in text material occur in the chapters on torsion, plastic deformation and mechanical properties of materials.

With regard to torsion, the problem of the twist of tubular members with intermediate cells is considered, as well as the torsional buckling of thin-walled members of open cross section. Each of these topics is important in the design of thin- walled structures such as the structural components of air- planes. In the chapter on plastic deformation the fundamental principles of limit design are discussed. Several examples of the application of the method to structural analysis are presented.

Major additions were made to the chapter on mechanical properties of materials, so that this single chapter now contains over 160 pages. The purpose of this expanded chapter is to focus attention on the recent developments in the field of experimental studies of the properties of structural materials. Some of the topics discussed are (1) the influence of imperfec- tions on the ultimate strength of brittle materials and the “size effect”; (2) comparison of test results for single-crystal and polycrystal specimens; (3) the testing of materials under two- and three-dimensional stress conditions and various strength theories; (4) the strength of materials under impact; (5) fatigue of metals under various stress conditions and methods for im- proving the fatigue resistance of machine parts; and (6) strength of materials at high temperature, creep phenomenon and the use of creep test data in design. For the reader who desires to expand his knowledge of these topics further, the numerous references to the recent literature will be helpful. Finally, in the concluding article of the book, information for the proper

. . . 111

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iv PREFACE TO THE THIRD EDITION

selection of working stresses is presented in considerable

detail.

It is the author’s hope that with these additions, the book

will be more complete for the teaching of graduate courses in

mechanics of materials and also more useful for designers and

research engineers in mechanical and structural engineering.

In conclusion the author wishes to thank Professor James M.

Gere of Stanford University for his assistance and numerous

suggestions in revising the book and in reading the proofs. S. TIMOSHENKO

STANFORD UNIVERSITY

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PREFACE TO THE SECOND EDITION

In the preparation of the new edition of this volume, the general character of the book has remained unchanged; the only effort being to make it more complete and up-to-date by including new theoretical and experimental material repre-

senting recent developments in the fields of stress analysis and experimental investigation of mechanical properties of structural materials.

The most important additions to the first edition include: I. A more complete discussion of problems dealing with bending, compression, and torsion of slender and thin-walled structures. This kind of structure finds at present a wide application in airplane constructions, and it was considered desirable to include in the new edition more problems from that field.

2. A chapter on plastic defor mations dealing with bending and torsion of beams and shafts beyond the elastic limit and also with plastic flow of material in thick-walled cylinders subjected to high internal pressures.

3. A considerable amount of new material of an experimental character pertaining to the behavior of structural

materials at high temperatures and to the fatigue of metals under reversal of stresses, especially in those cases where fatigue is combined with high stress concentration.

4. Important additions to be found in the portion of the book dealing with beams on elastic foundations; in the chapters on the theory of curved bars and theory of plates and shells; and in the chapter on stress concentration, in which some recent results of photoelastic tests have been included.

Since the appearance of the first edition of this book, the author’s three volumes of a more advanced character, “Theory of Elasticity,” “Th eory of Elastic Stability,” and “Theory of Plates and Shells” have been published. Reference to these

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vi PREFACE TO THE SECOND EDITION

books are made in various places in this volume, especially

in those cases where only final results are given without a

complete mathematical derivation.

It is hoped that with the additions mentioned above the

book will give an up-to-date presentation of the subject of

strength of materials which may be useful both to graduate

students interested in engineering mechanics and to design

engineers dealing with complicated problems of stress analysis.

PALO ALTO, CALIFORNIA June 12, 1941

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PREFACE TO THE FIRST EDITION

The second volume of

THE STRENGTH OF MATERIALS

is

written principally for advanced students, research engineers, and

designers. Th e writer has endeavored to prepare a book

which contains the new developments that are of practical importance in the fields of strength of materials and theory of elasticity. Complete derivations of problems of practical interest are given in most cases. In only a comparatively few cases of the more complicated problems, for which solutions cannot be derived without going beyond the limit of the usual standard in engineering mathematics, the final results only are given. In such cases, the practical applications of the results are discussed, and, at the same time, references are given to the literature in which the complete derivation of the solution can be found.

In the first chapter, more complicated problems of bending of prismatical bars are considered. The important problems of bending of bars on an elastic foundation are discussed in detail and applications of the theory in investigating stresses in rails and stresses in tubes are given. The application of trigonometric series in investigating problems of bending is also discussed, and important approximate formulas for

combined direct and transverse loading are derived.

In the second chapter, the theory of curved bars is developed in detail. The application of this theory to machine design is illustrated by an analysis of the stresses, for instance, in hooks, fly wheels, links of chains, piston rings, and curved pipes.

The third chapter contains the theory of bending of plates. The cases of deflection of plates to a cylindrical shape and the symmetrical bending of circular plates are discussed in detail and practical applications are given. Some data regarding the bending of rectangular plates under uniform load are also given.

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. . .

VI11 PREFACE TO THE FIRST EDITION

In the fourth chapter are discussed problems of stress distribution in parts having the form of a generated body and symmetrically loaded. These problems are especially important for designers of vessels submitted to internal pressure and of rotating machinery. Tensile and bending

stresses in thin-walled vessels,stresses in thick-walled cylinders, shrink-fit stresses, and also dynamic stresses produced in rotors and rotating discs by inertia forces and the stresses due to non-uniform heating are given attention.

The fifth chapter contains the theory of sidewise buckling of compressed members and thin plates due to elastic instability. Th ese problems are of utmost importance in many modern structures where the cross sectional dimensions are being reduced to a minimum due to the use of stronger materials and the desire to decrease weight. In many cases, failure of an engineering structure is to be attributed to elastic instability and not to lack of strength on the part of the material.

In the sixth chapter, the irregularities in stress distribution produced by sharp variations in cross sections of bars caused by holes and grooves are considered, and the practical significance of stress concentration is discussed. The photoelastic method, which has proved very useful in investigating stress concentration, is also described. The membrane analogy in torsional problems and its application in investigating stress concentration at reentrant corners, as in rolled sections and in tubular sections, is explained. Circular shafts of variable diameter are also discussed, and an electrical analogy is used in explaining local stresses at the fillets in such shafts.

In the last chapter, the mechanical properties of materials are discussed. Attention is directed to the general principles rather than to a description of established, standardized methods of testing materials and manipulating apparatus. The results of modern investigations of the mechanical properties of single crystals and the practical significance of this information are described. Such subjects as the fatigue of metals and the strength of metals at high temperature are

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PREFACE TO THE FIRST EDITION ix of decided practical interest in modern machine design. These problems are treated more particularly with reference to new developments in these fields.

In concluding, various strength theories are considered. The important subject of the relation of the theories to the method of establishing working stresses under various stress conditions is developed.

It was mentioned that the book was written partially for teaching purposes, and that it is intended also to be used for advanced courses. The writer has, in his experience, usually divided the content of the book into three courses as follows: (I) A course embodying chapters I, 3, and 5 principally for advanced students interested in structural engineering. (2) A course covering chapters 2, 3, 4, and 6 for students whose chief interest is in machine design. (3) A course using chapter 7 as a basis and accompanied by demonstrations in the material testing laboratory. The author feels that such a course, which treats the fundamentals of mechanical properties of materials and which establishes the relation between these properties and the working stresses used under various conditions in design, is of practical importance, and more attention should be given this sort of study in our engineering curricula.

The author takes this opportunity of thanking his friends who have assisted him by suggestions, reading of manuscript and proofs, particularly Messrs. W. M. Coates and L. H. Donnell, teachers of mathematics and mechanics in the Engineering College of the University of Michigan, and Mr. F. L. Everett of the Department of Engineering Research of the University of Michigan. He is indebted also to Mr. F. C. Wilharm for the preparation of drawings, to Mrs. E. D. Webster for the typing of the manuscript, and to the D. Van Nostrand Company for their care in the publication of the book.

S. TIMOSHENKO

ANN ARBOR, MICHIGAN

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NOTATIONS

a. ... Angle, coefficient of thermal expansion, numer-

ical coefficient

P ... Angle, numerical coefficient

Y ... Shearing strain, weight per unit volume A ... Unit volume expansion, distance

6. ... Total elongation, total deflection, distance e ... Unit strain

% %I> Ed ... Unit strains in x, y, and z directions

8. ... Angle, angle of twist per unit length of a shaft p ... Poisson’s ratio

p ... Distance, radius u ... Unit normal stress Ul, u2, u3. ... Principal stresses

*z> uy, u,. ... Unit normal stresses on planes perpendicular to the x, y, and z axes

UE. ... Unit stress at endurance limit ulllt. ... Ultimate stress

cr‘uC) uut. ... Ultimate stresses in compression and tension uw ... Working stress

c7‘y.p ... Yield point stress 7 ... Unit shear stress

TSY9 Tyz, TZX Unit shear stresses on planes perpendicular to the x, y, and z axes, and parallel to the y, z, and x axes

TE ... Endurance limit in shear

roct ... Unit shear stress on octahedral plane Tult. ... Ultimate shear stress

7-w. ... Working stress in shear 7y.p ... Yield point stress in shear $0. ... Angle, angle of twist of shaft w ... Angular velocity

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NOTATIONS xi

A. . . .

Cross-sectional area a, 6, c, d, e. . Distances c... Torsional rigidity C,. . . Warping rigidity D . . . . Flexural rigidity

d

Eye;; 2,: : : : Diameter

Modulus of elasticity, tangent modulus, reduced modulus

2,. ~ . . . . . . Shear flow

. . . . . . . . Modulus of elasticity in shear h.. . . . . . Height, thickness

I,,I, . . . . . . . Polar moments of inertia of a plane area with respect to centroid and shear center

I,,I,,I Z.... Moments of inertia of a plane area with respect to x, y, and z axes

k . . . . ...*... Modulus of foundation, radius of gyration, stress

concentration factor, numerical constant 2

ii.:::::::::

Length, span Bending moment

M&t. . . Ultimate bending moment

MY.p.. Bending moment at which yielding begins Mt. . . . Torque

(Mdulr. . . Ultimate torque

(Mt)Y.p.. . Torque at which yielding begins &.:::::::

Factor of safety Concentrated forces

p . . . . . . . Pressure, frequency of vibration

q... Load per unit length, reduction in area, sensi- tivity factor

R . . . . . . . . . . Reaction, force, radius, range of stress r . . . . . . . . . . . Radius, radius of curvature

S. . . . . Axial force, surface tension s . . . . . . . Length

T. . . Axial force, absolute temperature t u:::::::::: Temperature, thickness Strain energy u Ft.:: 1::::::

Rate of strain, displacement in x direction Volume, shearing force

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xii NOTATIONS

V . . . Velocity, creep rate, displacement in y direction

W.. . . . Weight

W . . . . . . . . . Strain energy per unit volume, displacement in

z direction

x,y,z . . . Rectangular coordinates 2. . . . . . . . . . Section modulus

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CONTENTS

CHAPTER PAGE

I. BEAMS ON ELASTIC FOUNDATIONS . . . 1

1. Beams of Unlimited Length . . . 1 2. Semi-infinite Beams . . . 11 3. Beams of Finite Length on Elastic Foundations . . 15

II. BEAMS WITH COMBINED AXIAL AND LATERAL LOADS . 26

4. Direct Compression and Lateral Load . . . 26 5. Continuous Struts . . . 37 6. Tie Rod with Lateral Loading . . . 41 7. Representation of the Deflection Curve by a Trig-

onometric Series . . . 46 8. Deflection of Bars with Small Initial Curvature . . 54 III. SPECIAL PROBLEMS IN THE BENDING OF BEAMS . . 57 9. Local Stresses in the Bending of Beams . . 57 10. Shearing Stresses in Beams of Variable Cross Section 62 11. Effective Width of Thin Flanges . . . 64 12. Limitations of the Method of Superposition . . . . 69

IV. THIN PLATES AND SHELLS . . . 76

13. Bending of a Plate to a Cylindrical Surface . . 76 14. Bending of a Long, Uniformly Loaded Rectangular

Plate. . . 78 15. Deflection of Long Rectangular Plates Having a

Small Initial Cylindrical Curvature . . . 84 16. Pure Bending in Two Perpendicular Directions . 86 17. Thermal Stresses in Plates . . . 90 18. Bending of Circular Plates Loaded Symmetrically

with Respect to the Center . . . 92 19. Bending of a Uniformly Loaded Circular Plate . 96 20. Bending of Circular Plates of Variable Thickness . 102 21. Bending of a Circular Plate Loaded at the Center . 103 22. Bending of a Circular Plate Concentrically Loaded 107 23. Deflection of a Symmetrically Loaded Circular Plate

with a Circular Hole at the Center . . . 109 24. Bending of Rectangular Plates . . . 114

. . . Xl,,

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xiv CONTENTS

CHAPTER PAGE

25. Thin-walled Vessels Subjected to Internal Pressure 117

26. Local Bending Stresses in Thin Vessels . . . 124

27. Thermal Stresses in Cylindrical She!ls . . . 134

28. Twisting of a Circular Ring by Couples Uniformly Distributed along Its Center Line . . . 138

V. BUCKLING OF BARS, PLATES AND SHELLS . . . . 29. Lateral Buckling of Prismatic Bars: Simpler Cases 30. Lateral Buckling of Prismatic Bars: More Com- plicated Cases . . . . 31. Energy Method of Calculating Critical Compressive Loads . . . . 32. Buckling of Prismatic Bars under the Action of Uni- formly Distributed Axial Forces . . . _ . . . . 33. Buckling of Bars of Variable Cross Section . . . . 34. Effect of Shearing Force on the Critical Load . . 35. Buckling of Latticed Struts . . . . _ . . . . 36. Inelastic Buckling of Straight Columns . . . . 37. Buckling of Circular Rings and Tubes under External Pressure . . . . 38. Buckling of Rectangular Plates . . _ 39. Buckling of Beams without Lateral Supports . . . 145 145 1.53 161 167 169 171 173 178 186 193 199 VI. DEFORMATIONS SYMMETRICAL ABOUT AN AXIS . . . . 205

40. Thick-walled Cylinder . . . 205

41. Stresses Produced by Shrink Fit . . . , . . . 210

42. Rotating Disc of Uniform Thickness . . . 214

43. Rotating Disc of Variable Thickness . . . 223

44. Thermal Stresses in a Long, Hollow Cylinder . . . 228 VII. TORSION . . . .

4.5. Shafts of Noncircular Cross Section . . . . 46. Membrane Analogy . . . . 47. Torsion of Rolled Profile Sections . . . . 48. Torsion of Thin Tubular Members . . . . . 49. Torsion of Thin-walled Members of Open Cross Sec-

tion in Which Some Cross Sections Are Prevented from Warping . . . . 50. Combined Bending and Torsion of Thin-walled Mem-

bers of Open Cross Section . . . . 51. Torsional Buckling of Thin-walled Members of Open

Cross Section . . . . 235 23.5 237 244 247 255 267 273

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CONTENTS xv

CHAPTER PAGE

52. Buckling of Thin-walled Members of Open Cross Section by Simultaneous Bending and Torsion . 279 53. Longitudinal Normal Stresses in Twisted Bars , 286

54. Open-coiled Helical Spring . . . 292

VIII. STRESS CONCENTRATION . . . . 55. Stress Concentration in Tension or Compression Members . . . . 56. Stresses in a Plate with a Circular Hole . . . . . 57. Other Cases of Stress Concentration in Tension Members . . . . 58. Stress Concentration in Torsion . . . . 59. Circular Shafts of Variable Diameter . . . . 60. Stress Concentration in Bending . . . . 61. The Investigation of Stress Concentration with Models . . . . 62. Photoelastic Method of Stress Measurements . . . 63. Contact Stresses in Balls and Rollers . . . . IX. DEFORMATIONS BEYOND THE ELASTIC LIMIT . . . . . 64. Structures of Perfectly Plastic Materials . . . . . 65. Ultimate Strength of Structures . . . . 66. Pure Bending of Beams of Material Which Does Not Follow Hooke’s Law . . . _ . . . . 67. Bending of Beams by Transverse Loads beyond the Elastic Limit . . . _ . 68. Residual Stresses Produced by Inelastic Bending _ 69. Torsion beyond the Elastic Limit . . . . 70. Plastic Deformation of Thick Cylinders under the Action of Internal Pressure . . . . 300 300 301 306 312 318 324 329 333 339 346 346 354 366 374 377 381 386 X. MECHANICAL PROPERTIES OF MATERIALS . . . 393

71. General . . . 393

72. Tensile Tests of Brittle Materials . . . , 395

73. Tensile Tests of Ductile Materials . . . 400

74. Tests of Single-Crystal Specimens in the Elastic Range . . . 403

7.5. Plastic Stretching of Single-Crystal Specimens . . 407

76. Tensile Tests of Mild Steel in the Elastic Range . 411 77. Yield Point . , . . . 417

78. Stretching of Steel beyond the Yield Point . . 420

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xvi CONTENTS

CHAPTER PAGE

80. Compression Tests . . . 435 81. Tests of Materials under Combined Stresses . 438 82. Strength Theories . . . . 444 83. Impact Tests . . . 462 84. Fatigue of Metals . . . 470 8.5. Fatigue under Combined Stresses . . . 479 86. Factors Affecting the Endurance Limit . . 483 87. Fatigue and Stress Concentrations . . . 489 88. Reduction of the Effect of Stress Concentrations in

Fatigue . . . 498

89. Surface Fatigue Failure . 505 90. Causes of Fatigue . . . , . . . 509 91. Mechanical Properties of Metals at High Tempera-

tures . . . 516 92. Bending of Beams at High Temperatures 527 93. Stress Relaxation . . . S30 94. Creep under Combined Stresses . . . 533 95. Particular Cases of Two-dimensional Creep . . 537

96. Working Stresses . . 544

AUTHOR INDEX . . . , . . 5.59 SUBJECT INDEX . . . 565

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PART

II

CHAPTER I

BEAMS ON ELASTIC FOUNDATIONS

1. Beams of Unlimited Length.-Let us consider a prismatic beam supported along its entire length by a continuous elastic foundation, such that when the beam is deflected, the intensity of the continuously distributed reaction at every point is proportional to the deflection at that point.1 Under such conditions the reaction per unit length of the beam can be represented by the expression ky, in which y is the deflection and k is a constant usually called the modulus ofthefoun-

dation. This constant denotes the reaction per unit length

when the deflection is equal to unity. The simple assumption that the continuous reaction of the foundation is proportional to the deflection is a satisfactory approximation in many practical cases. For instance, in the case of railway tracks the solution obtained on this assumption is in good agreement with actual measurements.2 In studying the deflection curve of the beam we use the differential equation 3

EI, $ = q,

₍₄

1 The beam is imbedded in a material capable of exerting downward as well as upward forces on it.

2 See S. Timoshenko and B. F. Langer, Trans. A.S.M.E., Vol. 54, p. 277, 1932. The theory of the bending of beams on an elastic foundation has been developed by E. Winkler, Die Lehre von der Elastizit& und Festigkeit, Prague, p. 182, 1867. See also H. Zimmermann, Die Berechnung des Eisenbahn-

Oberbaues, Berlin, 1888. Further development of the theory will be found in: Hayashi, Theorie des Triigers auf elastischer Unterlage, Berlin, 1921; Wieghardt, 2. angew. Math. U. Mech., Vol. 2, 1922; K. v. Sanden and Schleicher, Beton u. Eisen, Heft 5, 1926; Pasternak, Beton u. Eisen, Hefte 9 and 10, 1926; W. Prager, 2. angew. Math. u. Mech., Vol. 7, p. 354, 1927; M. A. Biot, J. Appl. Mech., Vol. 4, p. A-l, 1937; M. Hetenyi, Beams on

Elastic Foundation, Ann Arbor, 1946.

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2 STRENGTH OF MATERIALS

in which 4 denotes the intensity of the load acting on the beam. For an unloaded portion the only force on the beam is the con-

tinuously distributed reaction from the foundation of intensity ky. Hence q = -ky, and eq. (a) becomes

(2) the general solution of eq. (1) can be represented as follows: y

= P(A cos /3x + B

sin &v)

+ e+“(C cos /?x + II sin /3x). (b) This can easily be verified by substituting the value from eq. (6) into eq. (1). In particular cases the constants A,

B,

C and D of the solution must be determined from the known conditions at certain points.

Let us consider as an example the case of a single concentrated load acting on an infinitely long beam (Fig. la), taking the origin of coordinates at the point of application of the

force. Because of the condition of sym-

-r &P

////////w//I/// /, * metry, only that part of the beam to the

fo) 0 +L

right of the load need be considered (Fig. lb). In applying the general solution, /

f _.? /d eq. (b), to this case, the arbitrary constants must first be found. It is reason-

FIG. 1. able to assume that at points infinitely

distant from the force P the deflection and the curvature vanish. This condition can be fulfilled only if the constants A and

B

in eq. (6) are taken equal to zero. Hence the deflection curve for the right portion of the beam becomes

y = e+“(C cos @x + D sin px).

The two remaining constants of integration C and D are found from the conditions at the origin, x = 0. At this point

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BEAMS ON ELASTIC FOUNDATIONS 3

the deflection curve must have a horizontal tangent; therefore

d.

c-1

dx z=,,= 0,

or, substituting the value ofy from eq. (c),

e+(C cos fix + D sin px + C sin px - D cos /3~)~=s = 0,

from which

C = D.

Eq. (c) therefore becomes

y = Ce+(cos /3x + sin px).

The consecutive derivatives of this equation are

dr

z= -2fiCe+” sin fix,

d2r

- = 2p2Ce+“(sin @x - COS /3x),

dx2

d3r

- =

4p3CedsZ cos fix. dx3

(4

(f>

The constant C can now be determined from the fact that at

x = 0 the shearing force for the right part of the beam (Fig.

lb) is equal to - (P/2). Th e minus sign follows from our con-

vention for signs of shearing forces (see Part I, pp. 75-6).

Then

(y>z=o = (g) = = -EI, (2)

z 0 z-o

= _ f, or, using eq.

(f),

from which

EI,.4p3C = f,

P

c=---.

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Substituting this value into eqs. (d) and (e), we obtain the

following equations for the deflection and bending moment

curves :

P

~ e+(cos /3x + sin /3x)

’ = 8p3EI,

PP z----e

2k

+z(cos ,8x + sin /3x),

(3)

M = -EI, 2 = - $ e-@(sin /3x - cos px).

(4)

Eqs. (3) and (4) each have, when plotted, a wave form with

gradually diminishing amplitudes. The length a of these

0

0.4

l/l /I I I I I I I I I I

0 1 2 3 4 5

FIG. 2.

waves is given by the period of the functions cos /3x and sin /3x, i.e.,

a=2rr=2*4-.

J

4EI,

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To simplify the calculation of deflections, bending moments

and shearing forces a numerical table is given (Table l), in

TABLE 1: FUNCTIONS p, +, 13 AND r

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 rp 1.0000 0.9907 0.9651 0.9267 0.8784 0.8231 0.7628 0.6997 0.6354 0.5712 0.5083 0.4476 0.3899 0.3355 0.2849 0.2384 0.1959 0.1576 0.1234 0.0932 0.0667 0 0439 0.0244 0.0080 -0.0056 -0.0166 -0.0254 -0.0320 -0.0369 -0.0403 -0.0423 -0.0431 -0 0431 -0.0422 -0.0408 -0.0389 8 f 1.0000 l.OOOC 0 0.8100 0.9003 0.0903 0.6398 0.8024 0.1627 0.4888 0.7077 0.2189 0.3564 0.6174 0.2610 0.2415 0.5323 0.2908 0.1431 0.4530 0.3099 0.0599 0.3798 0.3199 -0.0093 0.3131 0.3223 -0.0657 0.2527 0.3185 -0.1108 0.1988 0.3096 -0.1457 0.1510 0.2967 -0.1716 0.1091 0.2807 -0.1897 0.0729 0.2626 -0.2011 0.0419 0.2430 -0.2068 0.0158 0.2226 -0.2077 -0.0059 0.2018 -0.2047 -0.0235 0.1812 -0.1985 -0.0376 0.1610 -0.1899 -0.0484 0.1415 -0.1794 -0.0563 0.1230 -0.1675 -0.0618 0.1057 -0.1548 -0.0652 0.0895 -0.1416 -0.0668 0.0748 -0.1282 -0.0669 0.0613 -0.1149 -0.0658 0.0492 -0.1019 -0.0636 0.0383 -0.0895 -0.0608 0.0287 -0.0777 -0.0573 0.0204 -0.0666 -0.0534 0.0132 -0.0563 -0.0493 0.0070 -0.0469 -0.0450 0.0019 -0.0383 -0.0407 -0.0024 -0.0306 -0.0364 -0.0058 -0.0237 -0.0323 -0.0085 -0.0177 -0.0283 -0.0106 0 e t - 3.6 -0.0366 -0.0124 -0.0245 -0.0121 3.7 -0.0341 -0.0079 -0.0210 -0.0131 3.8 -0.0314 -0.0040 -0.0177 -0.0137 3.9 -0.0286 -0.0008 -0.0147 -0.0140 4.0 -0.0258 0.0019 -0 0120 -0.0139 4.1 -0.0231 0.0040 -0.0095 -0.0136 4.2 -0.0204 0.0057 -0.0074 -0.0131 4.3 -0.0179 0.0070 -0.0054 -0.0125 4.4 -0.0155 0.0079 -0.0038 -0.0117 4.5 -0.0132 0.0085 -0 0023 -0.0108 4.6 -0.0111 0.0089 -0 0011 -0.0100 4.7 -0.0092 0.0090 0.0001 -0.0091 4.8 -0.0075 0.0089 0.0007 -0.0082 4.9 -0.0059 0.0087 0.0014 -0.0073 5.0 -0.0046 0.0084 0.0019 -0.0065 5.1 -0.0033 0.0080 0.0023 -0.0057 5.2 -0.0023 0.0075 0.0026 -0.0049 5.3 -0.0014 0.0069 0.0028 -0.0042 5.4 -0.0006 0.0064 0.0029 -0.0035 5.5 0.0000 0.0058 0.0029 -0.0029 5.6 0.0005 0.0052 0.0029 -0.0023 5.7 0.0010 0.0046 0.0028 -0.0018 5.8 0.0013 0.0041 0.0027 -0.0014 5.9 0.0015 0.0036 0.0026 -0.0010 6.0 0.0017 0.0031 0.0024 -0.0007 6.1 0.0018 0.0026 0.0022 -0.0004 6.2 0.0019 0.0022 0.0020 -0.0002 6.3 0.0019 0.0018 0.0018 tO.OOO1 6.4 0.0018 0.0015 0.0017 0.0003 6.5 0.0018 0.0012 0.0015 0.0004 6.6 0.0017 0.0009 0.0013 0.0005 6.7 0.0016 0.0006 0.0011 0.0006 6.8 0.0015 0.0004 0.0010 0.0006 6.9 0.0014 0.0002 0.0008 0.0006 7.0 0.0013 0.0001 0.0007 0.0006 -

which the following notations are used:

cp = e+z(cos px + sin /3x); $ = - e+z(sin fix - cos px);

8 = evpz cos /3x; r = empz sin fix.

In Fig. 2 the functions cp and $ are shown graphically.

(22)

Using notations (6) and eqs. (d)-(f), we obtain

y = gy P(PX>, p =

d2y

P

~4 = -EIz dx2 = G Mx),

Y= -EL2 = - ;s(fl,).

(7)

From these equations and Table 1, the deflection, slope, bend-

ing moment and shearing force for any cross section of the

beam can be readily calculated. The maximum deflection and

maximum bending moment occur at the origin and are, respec-

tively,

6 = (y)&J = $9

By using the solution (eq. 3) for a single load and the principle

of superposition, the deflection produced in an infinitely long

beam on an elastic foundation by any other type of loading

can be readily obtained.

As an example let us consider the case of a uniform load distributed over a length I of an infinitely long beam (Fig. 3). Consider any point A, and let c and b represent the dis-

FIG. 3. tances from this point to the ends of the loaded

part of the beam. The deflection at A, produced by an element qdx of the load, is obtained by substituting qdx for P in eq. (3), which gives

qdx

_ e-pz(cos /3x + sin /3x). 8p3 EI,

The deflection produced at A by the loading distributed over the length I then becomes

(23)

S

b qdx

Y” ~ edBz(cos px + sin /3x) o 8P3EIz

+5(cos fix + sin fix)

= & (2 - emflb cos /3S - emBc cos PC). ₍₉₎ If c and b are large, the values eWBb and e-s’ will be small and the deflection (eq. R) will be equal approximately to q/k; i.e., at points remote from the ends of the loaded part of the bar the bending of the bar can be neglected and it can be assumed that the uniform loading 4 is transmitted directly to the elastic foundation. Taking the point A at the end of the loaded part of the bar, we have c = 0, b = I, e-s’ cos @c = 1. Assuming that I is large, we have also e-fib cos fib = 0. Then y = q/2k; i.e., the deflection now has only one-half of the value obtained above.

In a similar manner, by using eq. (4), the expression for the bending moment at A can be derived. If the point A is taken outside the loaded portion of the beam and if the quantities b and c represent, respectively, the larger and the smaller distance from this point to the ends of the loaded part of the beam, the deflection at A is

S

b qdx

Y= ___ e-@ (cos px + sin @x) 0 8P3EIz

S

c qdx

- _{-___} _{eeBz (cos /3x + sin px)} o W3EI,

= -& (ep8” cos @c - emBb cos fib).

When c = 0 and if b = I is a large quantity, we obtain for the deflection the value q/2k, which coincides with our previous conclusion. As the distances b and c increase, the deflection, eq. (h), decreases, approaching zero as b and c grow larger.

-x C-b) The case of a couple acting

on an infinitely long beam, Fig. FIG. 4.

4~2, can also be analyzed by using the solution, eq. (3), for a single load. The action of the couple is equivalent to that of the two forces P

(24)

STRENGTH OF MATERIALS

shown in Fig. 4b, if Pe approaches MO while e approaches zero. Using the first of eqs. (7), we find the deflection at a distance x from the origin:

y = g (CP@X)

- dP(x + e>ll

MOP dP”4 - dP(x + e>l

MOP

&

=-.

=--

2k e 2k z’

From eqs. (7),

and the deflection curve produced by the couple MO becomes

By differentiating this equation, we obtain

dr

MoP3

- -wx>, i&- k

d”y MO

M = - EI, dx2 = T 0(0x),

_(10’)

Using these equations together with Table 1, we can readily calculate the deflection, slope, bending moment and shearing force for any cross section of the beam.

We shall now consider the case of several loads acting on an infinite beam. As an example, bending of a rail produced by the wheel pressures of a locomotive will be discussed. The following method of analyzing stresses in rails is based upon the assumption that there is a continuous elastic support under the rail. This assumption is a good approximation,4 since the distance between the ties is small in comparison to the wavelength a of the deflection curve, given by eq. (5). I n order to obtain the magnitude k of the modulus of the foundation, the load required to produce unit deflection of a tie must be divided by the tie spacing. It is assumed that the tie is

4 See the author’s paper, “Strength of Rails,” Memoirs Inst. Engrs. Ways of Communication (St. Petersburg), 1915; and the author’s paper in Proc. Zd Internat. Congr. Appl. Mech., Ziirich, 1926. See also footnote 2.

(25)

BEAMS ON ELASTIC FOUNDATIONS

symmetrically loaded by two loads corresponding to the rail pressures. Suppose, for example, that the tie is depressed 0.3 in. under each of two loads of 10,000 lb and that the tie spacing is 22 in.; then

k= 10,000

0.3 x 22 = 1,500 lb per sq in.

For the case of a single wheel load P, eqs. (8) and (9) are used for the maximum deflection and maximum bending moment. The maximum stress due to the bending of the rail will be

where 2 denotes the section modulus of the rail.5

In order to compare the stresses in rails which have geometrically similar cross sections, eq. (i) may be put in the following form:

in which A is the area of the cross section of the rail. Since the second factor on the right-hand side of eq. (j) remains constant for geometrically similar cross sections and since the third factor does not depend on the dimensions of the rail, the maximum stress is inversely proportional to the area of the cross section, i.e., inversely proportional to the weight of the rail per unit length.

An approximate value of the maximum pressure R,,, on a tie is obtained by multiplying the maximum depression by the tie spacing I and by the modulus of the foundation. Thus, using eq. (S), we have

‘“/k++. J

ki4 R

max = 2k 4EI,

It may be seen from this that the pressure on the tie depends principally on the tie spacing 1. It should also be noted that k occurs in both eqs. (j) and (k) as a fourth root. Hence an error in the determination of k will introduce only a much smaller error in the magnitude of urnax and R,,,.

6 In writing eq. (i) it was assumed that the elementary beam formula can be used at the cross section where the load P is applied. More detailed investigations show that because of local stresses, considerable deviation from the elementary eq. (i) may be expected.

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When several loads are acting on the rail, the method of superposition must be used. To illustrate the method of calculation we shall discuss a numerical example. Consider a loo-lb rail section with .l, = 44 in.4 and with a tie spacing such that k = 1,500 lb per sq in.; then from eq. (2)

and from eq. (5)

a = 2 = 272 in. P

We take as an example a system of four equal wheel loads, 66 in. apart. If we fix the origin of coordinates at the point of contact of the first wheel, the values of px for the other wheels will be those given in Table 2. Also given are the corresponding values of the functions (p and #, taken from Table 1.

TABLE 2 Loads 1 2 3 4 -- /3x. . . 0 1.52 3.05 4.57 I). . . 1 -0.207 -0.051 0.008 ‘p... 1 0.230 -0.042 -0.012

After superposing the effects of all four loads acting on the rail, the bending moment under the first wheel is, from eq. (4),

Ml = $ (1 - 0.207 - 0.051 + 0.008) = 0.75 $t

i.e., the bending moment is 25 per cent less than that produced by a single load P. Proceeding in the same manner for the point of contact of the second wheel we obtain

A42

= 50

- 2 x 0.207 - 0.051) = 0.535 ;-

It may be seen that owing to the action of adjacent wheels the bending moment under the second wheel is much smaller than that under

(27)

the first. This fact was proved by numerous experimental measurements of track stresses.

Using eq. (3) and the values in the last line of Table 2, we find the following deflection under the first wheel:

aI = g (1 + 0.230 - 0.042 - 0.012) = 1.18;.

The deflections at other points can be obtained in a similar manner. It is seen that the method of superposition may be easily applied to determine the bending of a rail produced by a combination of loads having any arrangement and any spacing.

The above analysis is based on the assumption that the rail support is capable of developing negative reactions. Since there is usually play between the rail and the spikes, there is little resistance to the upward movement of the rail, and this tends to increase the bending moment in the rail under the first and the last wheels. Never- theless, in general the above theory for the bending of a rail by static loading is in satisfactory agreement with the experiments which have been made.

Problems

1. Using the information given in Table 2, construct the bending moment diagram for a rail, assuming that the wheel pressures are equal to 40,000 lb. Such a diagram should show that the moments are negative in sections midway between the wheels, indicating that during locomotive motion the rail is subjected to the action of reversal of bending stresses, which may finally result in fatigue cracks. 2. Find the bending moment at the middle of the loaded portion of the beam shown in Fig. 3 and

the slope of the deflection curve at the left end of the same portion.

3. Find the deflection at any point A under the triangular load acting on an infinitely long beam on an elastic foundation, Fig. 5.

Answer. Proceeding as in the derivation of eq. (g), p. 7, we ob-

tain FIG. 5.

%J = -&; W(Pc> - m4 - w3w) + 4Pcl.

2. Semi-infinite Beams.-If a long beam on an elastic foundation is bent by a force

P

and a moment A40 applied at the end as shown

(28)

in Fig. 6, we can again use the general solution, eq. (6), of the preced- Since the deflection and the bending moment approach zero as the distance x from the loaded end increases, we must take A = B = 0 in that solution. We obtain FIG. 6. _{y = emBz(C cos px + D sin px).} _(a) For determining the constants of integration C and D we have the conditions at the origin, i.e., under the load P:

EI,$

_{( >}

=-A&,

CC-0 d"r E1z 2 ( > = -y,p. X=0

Substituting from eq. (a) into these equations, we obtain two linear equations in C and D, from which

c=

DC-.

MO

2p2EI,

Substituting into eq. (a), we obtain

e-Bz

’ = 2p3EI, ___ [P cos px - pMo(cos Ox - sin Px)]

or, using notations (6),

To get the deflection under the load we must substitute x = 0 into eq. (11). Then

6 = (y)z=o = & (P - PMO). (11’) .?

The expression for the slope is obtained by differentiating eq. (11). At the end (x = 0) this becomes

dr

0

% z=. = - 2p2EI, 1 (P - 2pM,). (12)

By using eqs. (II’) and (12) in conjunction with the principle of superposition, more complicated problems can be solved. Take as an example a uniformly loaded long beam on an elastic foundation,

(29)

BEAMS ON ELASTIC FOUNDATIONS ₁₃

having a simply supported end, Fig. 7~. The reaction R at the end is found from the condition that the deflection at the support is zero. Observing that at a large dis-

tance from the support, the bending of the beam is negligible, and that its depression into the foundation can be taken equal to g/k, we calculate the value of R by substituting MO = 0 and 6 = q/k into eq. (11’). This yields the result

R = 2p3EI, . f = 5. (13)

The deflection curve is now ob- _{I y} tained by subtracting the de-

flections given by eq. (11) for FIG. 7.

P = R, MO = 0 from the uniform depression q/k of the beam, which gives

Q e-Bz

v=-- _{____} _{R cos /3x = i} _{(1 - e--g’ cos px).}

k 2f13EI, (14)

In the case of a built-in end, Fig. 76, the magnitudes of the reaction R and of the moment MO are obtained from the conditions that at the support the deflection and the slope are zero. Observing that at a large distance from the support the deflection is equal to q/k and using eqs. (11’) and (12), we obtain the following equations 6 for calculating R and MO:

and 4 --= k - & _CTCR + MO) from which o= &- CR + WMo) z

The minus sign of MO indicates that the moment has the direction shown by the arrow at the left in Fig. 7b.

6 In eqs. (11’) and (12), P = --R is substituted, since the positive direction for the reaction is taken upwards.

(30)

1. Find the deflection tic foundation hinged at

curve for a semi-infinite beam on an elas- the end and acted upon by a couple Ma,

Fig. 8.

Solution. The reaction at the -x hinge is obtained from eq. (11’) by

substituting 6 = 0, which gives

Problems

FIG. 8. P = pi&.

Substituting this value of P in eq. (11) we obtain

Mo -82

Y=7_82EI,e sin px = & !xm. By subsequent differentiation, we find

dr

w31Mo

-=

dx

-

k

. ww,

M = --ELd$ = .%f,.e(px), I

J’ = -M, $ = -~Mo.cp(~x).

₁

(16)

(6

2. Find the bending moment MO and the force P acting on the end of a semi-infinite beam on an elastic foundation, Fig. 9, if the deflection 6 and the slope LY at the end are given.

I,

FIG. 9.

Solution. The values MO and P are obtained from eqs. (11’) and (12) by substituting the given quantities for 6 and (dy/dx),,o = a.

3. Find the deflection curve for a semi-infinite beam on an elastic foundation produced by a load P applied at a distance c from the free end A of the beam, Fig. 10.

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Solution. Assume that the beam extends to the left of the end A as shown by the dotted line. In such a case eq. (3) gives the deflection curve for x > 0, and at

the cross section A of the fictitious infinite beam we have, from eqs.

(7), and using the condition of sym- - --Y metry,

A4 = J&%), Y = $3c). (c) Y

To obtain the required deflection --x curve for the semi-infinite beam

free at the end A, we evidently (bl must superpose the deflection of

the semi-infinite beam produced FIG. 10.

by the forces shown in Fig lob on

the deflection of the fictitious infinite beam. By using equations (3), (11) and (c) in this way we obtain for x > 0:

+ wo[P(x

+ c>l - 3iwMP(x + c)l I.

(4

This expression can also be used for -c < x < 0; in this case we have only to substitute the absolute value of X, instead of x, in &?x).

3. Beams of Finite Length on Elastic Foundations.-The bending of a beam of finite length on an elastic foundation can also be inves- tigated by using the solution, eq. (3), for an infinitely long beam together with the method of superposition.7 To illustrate the method let us consider the case of a beam of finite length with free ends which is loaded by two symmetrically applied forces

P,

Fig. lla. A similar condition exists in the case of a tie under the action of rail pressures. To each of the three portions of the beam the general solution, eq. (6) of Art. 1, can be applied, and the constants of integration can be calculated from the conditions at the ends and at the points of application of the loads. The required solution can, however, be

7 This method of analysis was developed by M. Hetknyi, Final Report, Zd Congr. Internat. Assoc. Bridge and Structural Engng., Berlin, 1938. See also his Beams on Elastic Foundation, p. 38.

(32)

obtained much more easily by superposing the solutions for the two kinds of loading of an infinitely long beam shown in Fig. lib and c. In Fig. lib the two forces P are acting on an infinitely long beam. In Fig. llc the infinitely long beam is loaded by forces Qo and moments A&, both applied outside the portion AB of the beam, but infinitely close to points A and B which correspond to the free ends of the given beam, Fig. lla. It is easy to see that by a proper selection of the forces Q0 and the moments Ma, the bending moment and

FIG. 11.

the shearing force produced by the forces P at the cross sections A and B of the infinite beam (shown in Fig. 116) can be made equal to zero. Then the middle portion of the infinite beam will evidently be in the same condition as the finite beam represented in Fig. lla, and all necessary information regarding bending of the latter beam will be obtained by superposing the cases shown in Figs. 116 and 11~.

To establish the equations for determining the proper values of Ma and Qe, let us consider the cross section A of the infinitely long beam. Taking the origin of the coordinates at this point and using eqs. (7), the bending moment A4’ and the shearing force V produced at this point by the two forces P, Fig. 116, are

M’ = $ ~Gkw

- c>l

+ W>l-

Y’ = f {ew - c)] + e(pc>}

1

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The moment A4” and the shearing force VI’ produced at the same point by the forces shown in Fig. llc are obtained by using eqs. (7) together with eqs. (lo’), which give

M” = 2 [l +

vqPZ)l

+ y [I + W)l,

Qo

MOP

I

y” = - y I1 - fY(@l)J - 2 [l -

cp(Pl)l.

(4

The proper values of iI40 and Qo are now obtained from the equations M’ + M” = 0,

Yl + Y” = 0, (cl

which can be readily solved in each particular case by using Table 1. Once MO and Qo are known, the deflection and the bending moment at any cross se-&on of the actual

beam, Fig. 1 la, can be obtained by using eqs. (7), (10) and (10’) together with the method of superposition.

The particular case shown in Fig. 12 is obtained from our pre-

vious discussion by taking c = 0. FIG. 12.

Proceeding as previously explained, we obtain for the deflections at the ends and at the middle the following expressions:

2PP cash Pl + cos p(

ya =yb = T

sinh @l-I- sin ,N’ (4

cod1 p cos tf

4PP 2 2

yc = k sinh pl+ sin 81’ The bending moment at the middle is

sinh Esin 8’ MC = - !!! 2 2 p sinh /3I + sin pl’

(4

(34)

STRENGTH OF MATERIALS

The case of a single load at the middle, Fig. 13, can also be

X _{obtained from the previous case,}

shown in Fig. 11~. It is only necessary to take c = Z/2 and

FIG. 13. to substitute P for 2P. In this

way we obtain for the deflections at the middle and at the ends the following expressions:

cash p cos p 2Pp . 2 2 ya = yb = __ k sinh @+ sin pl’ (g) Pp cod-l /3l f cos pl f 2 Yc = z * sinh pl f sin pi For the bending moment under the load we find

P cash /?l - cos /3l A4, = -

4p sinh /9-l- sin pl’

FIG. 14.

(A)

The method used for the symmetrical case shown in Fig. lla can also be applied in the antisymmetrical case shown in Fig. 14~2. QO and MO in this case will also represent an antisymmetrical system as shown in Fig. 146. For the determination of the proper values of

(35)

BEAMS ON ELASTIC FOUNDATIONS ₁₉

QO and MO, a system of equations similar to eqs. (a), (6) and (c) can be readily written. As soon as Qo and A&, are calculated, all necessary information regarding the bending of the beam shown in Fig. 14a can be obtained by superposing the cases shown in Figures 146 and 14~.

Having the solutions for the symmetrical and for the antisymmetrical loading of a beam, we can readily obtain the solution for any kind of loading by using the principle of superposition. For example, the solution of the unsymmetrical case shown in Fig. 150 is obtained by superposing the solutions of the symmetrical and the antisymmetrical cases shown in Fig. 19 and c. The problem shown

FIG. 15. FIG. 16.

in Fig. 16 can be treated in the same manner. In each case the problem is reduced to the determination of the proper values of the forces QO and moments h/r, from the two eqs. (c).

In discussing the bending of beams of finite length we note that the action of forces applied at one end of the beam on the deflection at the other end depends on the magnitude of the quantity pl. This quantity increases with the increase of the length of the beam. At the same time, as may be seen from Table 1, the functions cp, $J and o are rapidly decreasing, and beyond a certain value of pl we can assume that the force acting at one end of the beam has only a negligible effect at the other end. This justifies our considering the beam as an infinitely long one. In such a case the quantities cp@Z), $(pl) and @(pl) can be neglected in comparison with unity in eqs. (b); by so doing eqs. (c) are considerably simplified.

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In general, a discussion of the bending of beams of finite length falls naturally into the three groups:

I. Short beams, pl < 0.60.

II. Beams of medium length, 0.60 < PZ < 5. III. Long beams, flZ > 5.

In discussing beams of group I we can entirely neglect bending and consider these beams as absolutely rigid, since the deflection due to bending is usually negligibly small in comparison with the deflection of the foundation. Taking, for example, the case of a load at the middle, Fig. 13, and assuming pl = 0.60, we find from the formulas given above for ya and yC that the difference between the deflection at the middle and the deflection at the end is only about one- half of one per cent of the total deflection. This indicates that the deflection of the foundation is obtained with very good accuracy by treating the beam as infinitely rigid and by using for the deflection the formula

The characteristic of beams of group II is that a force acting on one end of the beam produces a considerable effect at the other end. Thus such beams must be treated as beams of finite length.

In beams of group III we can assume in investigating one end of the beam that the other end is infinitely far away. Hence the beam can be considered as infinitely long.

In the preceding discussion it has been assumed that the beam is supported by a continuous elastic foundation, but the results obtained can also be applied when the beam is supported by a large number of equidistant elastic supports. As an example of this kind, let us consider a horizontal beam AB, Fig. 17, supporting a system of equidistant vertical beams which are carrying a uniformly distributed load q.8 All beams are simply supported at the ends. De- noting by EIl and Ii the flexural rigidity and the length of the vertical beams, we find the deflection at their middle to be

(j>

where R is the pressure on the horizontal beam AB of the vertical 8 Various problems of this kind are encountered in ship structures. A very complete discussion of such problems is given by I. G. Boobnov in his

Theory of Structure of Ships, St. Petersburg, Vol. 2, 1914. See also P. F. Pap- kovitch, Structural Mechanics of Ships, Moscow, Vol. 2, Part 1, pp. 318-814,

(37)

BEAMS ON ELA4STIC FOUNDATIONS

21

beam under consideration. Solving eq. (j) for R, we find that the horizontal beam AB is under the action of a concentrated force, Fig. 176, the magnitude of which is

5 48E11

R = s q11 - - 113 y-

(k)

Assuming that the distance a between the vertical beams is small in comparison with the length I of the horizontal beam and replacing

FIG. 17.

the concentrated forces by the equivalent uniform load, as shown in Fig. 17c, we also replace the stepwise load distribution (indicated in the figure by the broken lines) by a continuous load distribution of

the intensity

ql - b, where

5 Q/l 48E11

41=--i _{8 a} k=-. _all3 (4

The differential equation of the deflection curve for the beam AB

then is

EI> = q1 - ky.

(38)

loaded beam on an elastic foundation. The intensity of the load and the modulus of the foundation are given by eqs. (I).

In discussing the deflection of the beam we can use the method of superposition previously explained or we can directly integrate eq. (m). Using the latter method, we may write the general integral of eq. (m) in the following form:

y = T + C1 sin fix sinh /3x + Cs sin px cash /3x

+ Cs cos PX sinh /3x + C4 cos /?x cash fix. (n) Taking the origin of the coordinates at the middle, Fig. 17c, we con- clude from the condition of symmetry that

c, = c3 = 0.

Substituting this into eq. (n) and using the conditions at the simply supported ends,

(y)ez,2 = 0, = 0,

we find 2=112

2 sin ?sinh p_’ c ₁ =-!A 2 2

k cos fil+ cash fll’ 2 cot p’ cash p’

c=-Q’ ‘2 2

4

k cos 01 + cash /3l’ The deflection curve then is

2 sin e sinh p 2’ 2

cos fll+ cash /3l sin px sinh @x 2 cos p’ cash p_’

2 2

- COSPX coshj% .

cos ~1 f cash pl

(u)

The deflection at the middle is obtained by substituting x = 0, which gives

(39)

Substituting this value into eq. (k), we find the reaction at the middle support of the vertical beam, which intersects the beam AB at its mid-point. It is interesting to note that this reaction may be- come negative, which indicates that the horizontal beam actually supports the vertical beams only if it is sufficiently rigid; otherwise it may increase the bending of some of the vertical beams.

Problems

1. Find a general expression for the deflection curve for the beam illustrated in Fig. 12.

Answer.

2P/3 cash /3x cos /3(l - x) + cash /3(1 - x) cos px Y=T _{sinh pl + sin PI}

2. Find the deflections at the ends and the bending moment at the middle of the beam bent by two equal and opposite couples Ma, Fig. 18. FIG. 18. FIG. 19. hSW.??-. 2M,# sinh @I - sin pl Ya = Yb = - ~ k sinh pl + sin ~1’ sinh p cos e + cash e sin e

2 2 2 2

MC = 2Mo

’ sinh @I+ sin pl

3. Find the deflection and the bending moment at the middle of the beam with hinged ends, Fig. 19. The load P is applied at the middle of the beam.

Answer. _{Pp sinh @ - sin @l} ye = 2kcosh@+ cos ,H’ P sinh fil+ sin ~1 MC=--

(40)

24 STRENGTH OF M~4TERIALS

4. Find the deflection and the bending moment at the middle of the uniformly loaded beam with hinged ends, Fig. 20.

Answer.

pg

(

l-

sinh pl sin p’ &f, = z 2’ 2

p2 cash PI + cos /?l

5. Find the bending moments at the ends of the beam with built- in ends, carrying a uniform load and a load at the middle, Fig. 21.

FIG. 20. FIG. 21.

Answer.

sinh e sin ‘!?

&fo= -P 2 2 q sinh PI - sin pl fl sinh p1+ sin pi 2p2 sinh @+ sin ~1’ 6. Find the deflection curve for the beam on an elastic foundation with a load applied at one end, Fig. 22.

P Y FIG. 22. FIG. 23. Answer. 2PP *

’ = k(sinh2 pl - sin” 01) [sinh p/ cos fix cash p(l - X)

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BEAMS ON ELASTIC FOUNDATIONS 25 7. A beam on an elastic foundation and with hinged ends is bent by a couple MO applied at the end, Fig. 23. Find the deflection curve of the beam.

Answer.

211/r,p2

’ = k(cosh2 /3 - cos2 p(> [cash pl sin /3x sinh ~(1 - x)

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CHAPTER II

BEAMS WITH COMBINED AXIAL AND LATERAL LOADS

4. Direct Compression and Lateral Load.-Let us begin

with the simple problem of a strut with hinged ends, loaded

by a single lateral force P and centrally compressed by two

equal and opposite forces S, Fig. 24. Assuming that the strut

I,“’

FIG. 24.

has a plane of symmetry and that the force

P

acts in that

plane, we see that bending proceeds in the same plane. The

differential equations of the deflection curve for the two por-

tions of the strut are

Using the notation

s

EI = P2,

(a)

(17)

we represent the solutions of eqs. (a) and (b) in the following

form :

y=C,cospx+C,sinpx-gx, cc>

y = C3 cospx + C4 sinpx -

P(I - c)

sz (1 - x>. (4

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COMBINED AXIAL AND LATERAL LOADS 27

Since the deflections vanish at the ends of the strut, we con-

clude that

c, = 0,

Cs = -C4 tan pl.

The remaining two constants of integration are found from

the conditions of continuity at the point of application of the

load P, which require that eqs. (c) and (d) give the same de-

flection and the same slope for x = 2 - c. In this way we

obtain

C, sin p(l - c) = C,[sin p(l - c) - tan pl cosp(l - c)], c,p cos p(Z - c) = C,p[cos p(l - c) + tan pl sin p(l - c)] + g7 from which cz = P sin pc Sp sin pl’ c 4 = _ P sin PU - 4 ’ Sp tan pl

Substituting in eq. (c), we obtain for the left portion of the

strut

P sinpc . PC

’ = Sp sin pl sm px - SI xJ (18)

and by differentiation we find

4 P sin pc PC

dx= S sin pl cos px - SI’

d2r Pp sin pc

-=-

dx2 S sin pl sin px.

09)

The corresponding expressions for the right portion of the

strut are obtained by substituting (I - x) instead of x, and

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(18) and (19). Th ese substitutions give

P

sin p(l - c)

P(I - c)

Y=

Sp sin pl sin p(l - x) - sz (I - x), (20) d.

P

sin p(l - C)

P(I - 6)

-=-

dx

S sin pl

cos p(l - x) +

sz

’

(21) d2r

Pp

sin p(l - c) -=-

dx2

S sin pl sin p(Z - x). (22)

In the particular case when the load

P

is applied at the middle, we have c = 1/2, and by introducing the notation

we obtain from eq. (18)

(y>mx = (y)d,2 = & (tang - $7

PP

tan u - u =-.

48EI

&3

(23)

(24) The first factor in eq. (24) represents the deflection produced by the lateral load

P

acting alone. The second factor indicates in what proportion the deflection produced by

P

is mag- nified by the axial compressive force S. When S is small in comparison with the Euler load (S, =

EIT’/~~),

the quantity u is small and the second factor in eq. (24) approaches unity, which indicates that under this condition the effect on the deflection of the axial compressive force is negligible. When S approaches the Euler value, the quantity u approaches the value 7r/2 (see eq. 23) and the second factor in eq. (24) increases indefinitely, as should be expected from our previous discussion of critical loads (see Part I, p. 263).

The maximum value of the bending moment is under the load, and its value is obtained from the second of eqs. (19), which gives

M max

= E&,,?!=P(

. ~.

tanu

(25)

2=1/2

2s

2

4

2.4

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COMBINED AXIAL AND LATERAL LOADS 29

Again we see that the first factor in eq. (25) represents the

bending moment produced by the load P acting alone, while

the second factor is the magnzj2ation factor, representing the

effect of the axial force S on the maximum bending moment.

Having solved the problem for one lateral load

P,

Fig. 24,

we can readily obtain the solution for the case of a strut bent

by a couple applied at the end, Fig. 2.5. It is only necessary

Y

FIG. 25.

to assume that in our previous discussion the distance c is in-

definitely diminishing and approaching zero, while

PC

remains a

constant equal to MO. Substituting

PC = MO

and sin pc = pc

in eq. (18), we obtain the deflection curve

from which

MO sin px x

y=s ( - sin pl ---, I > (26)

The slopes of the beam at the ends are

1

=-. ~.

tan 2~ 1